CHAPTER VIII.

MOTION IN A RESISTING MEDIUM.

214. WHEN a body moves in a fluid, whether liquid or gaseous, it must, in displacing the particles of the medium and in rubbing against them, lose part of its own velocity. The resistance of a fluid to a body moving in it is therefore of the nature of a retarding force; but, in consequence of the great difficulty of making accurate experiments on the subject, the laws of the resistance of fluids have not yet been satisfactorily ascertained.

For a velocity neither very great nor very small, the general approximate law seems to be, that the resistance to a plane surface, moving with its plane at right angles to the line of motion, is proportional to the extent of the surface, the density of the resisting medium, and the square of the velocity taken conjointly. We are, however, only treating of the motion of a particle, in which the extent of surface has no place in our consideration, and will assume that the resistance depends entirely on the density of the medium and the velocity of the particle; illustrating, by supposing different laws, the method of procedure in all such cases.

215. A particle acted on by no forces is projected in a resisting medium of uniform density, of which the resistance varies as the nth power of the velocity; to determine the motion.

The motion will evidently be rectilinear. Let x be the distance of the particle from a fixed point in the line of motion at the time t, v its velocity at that time. The force due to the resistance may be represented by ko", k being a constant, and the equation of motion is

Suppose the particle to be projected from the origin, with velocity V, then when

the relation between v and t. It shews that v can never be zero if n > 1, but if n < 1 the velocity will become zero when

(1-n) k

at rest.

After this the particle will evidently remain

To find the distance of the particle from the origin at any time, we have from (2)

x and t being supposed to vanish together.

2n

1-7

When n<1, the distance to which the particle will go, or its distance from the

origin at the time

V1-n
(1 − n) k '

is

216. There is one case in which the above solution fails, namely when n=1, or the resistance varies as the velocity. In this case, by (1),

the constant being determined as before that x and t may vanish together.

Equations (3) and (4) determine the velocity and the position of the particle at any instant. They shew that the velocity continually diminishes without ever actually becoming zero, but that the space passed over by the particle can never be greater than a certain quantity, for when

217. A particle, acted on by a constant force in its line of motion, moves in a resisting medium of uniform density, of which the resistance varies as the square of the velocity; to determine the motion.

Suppose the particle projected from the origin with the velocity V, and let v be its velocity at any time t, x its distance from the origin at that time, and f the constant acceleration due to the force.

Assume K to be the velocity with which the particle would have to be animated that the resisting force might be equal to f, then the retarding force at any time may be reprev2

sented by f K

I. Let f act so as to diminish x; then the equation of motion is

Let T be the time at which the velocity becomes zero, and h the corresponding value of x, then

After this the particle begins to return, the force of resist

ance therefore tends to increase x, and the equation of

Let U be the velocity with which the particle will return to the point of projection; then, putting x=0 in the latter equation, we obtain